Page 244 - Fundamentals of Reservoir Engineering

OILWELL TESTING 182

8
As indicated by Cobb and Dowdle , equ. (7.40) can be solved for p D (t D) as
4t
t
p D () = 2 t DA + 1 2 ln D − 1 2 p D(MBH) ( DA ) (7.42)
t
π
D
γ
in which
4kh *
π
p D(MBH) ( DA ) = ( p − p )
t
qµ
is the dimensionless MBH pressure, which is simply the ordinate of the MBH chart
evaluated for the dimensionless flowing time t DA.

Equation (7.42) is extremely important since it represents the constant terminal rate
solution of the diffusivity equation which, for the case of a well draining from the centre
of a bounded, circular part of a reservoir, replaces the extremely complex form of
equ. (7.34). It should be noted, however, that the mathematical complexity of
equ. (7.34) is not being avoided since it is implicitly included in the MBH charts which
were evaluated using the method of images. Furthermore, equ. (7.42) is not restricted
to circular geometry and can be used for the range of the geometries and well
asymmetries included in the MBH charts.

For very short values of the flowing time t, when transient conditions prevail, the left
hand side of equ. (7.42) can be evaluated using equ. (7.23) and the former can be
reduced to

4kh *
π
)
p D(MBH) ( DA ) = ( p − p = 4 t DA (7.43)
t
π
qµ
Alternatively, for very long flowing times, when semi-steady state conditions prevail, the
left hand side of equ. (7.42) can be expressed as equ. (7.27) and in this case
equ. (7.42) becomes

4kh * r 2
π
)
t
p D(MBH) ( DA ) = ( p − p = ln C t w = ln (C t ) (7.44)
A D
A DA
qµ A
Inspection of the MBH plots of fig. 7.11, for a well situated at the centre of a regular
shaped bounded area, illustrates the significance of equs. (7.43) and (7.44). For small
values of the dimensionless flowing time t DA the semi-log plot of p D(MBH) vs t DA is non-
linear while for large t DA the plots are all linear as predicted by equ. (7.44), and have
unit slope (dp D(MBH)/d (In t DA) = 1). This latter feature is common to all the MBH charts,
figs. 7.11-15. that in each case there is a value of t DA, the magnitude of which depends
on the geometry and well asymmetry, for which the plots become linear indicating the
start of the semi-steady state flow condition. Furthermore, for the symmetry conditions
of fig. 7.11 there is a fairly sharp transition between transient and semi-steady state
flow at a value of t DA ≈ 0.1, which confirms the conclusion reached in sec. 7.4 and
exercise 7.4. For the geometries and various degrees of well asymmetry depicted in
the remaining charts, however, there is frequently a pronounced degree of curvature
extending to quite large values of t DA before the start of semi-steady state flow. This